Verified Mono-Monostatic Bodies
A mono-monostatic body is a homogeneous convex solid with exactly one stable and one unstable equilibrium (an equilibrium count, ECS, of 1) — the property that makes the Gömböc self-righting. This page is the canonical reference for a computationally verified thirteen-member catalog, spanning two construction families, with every member confirmed at ECS = 1. The full meshes and analysis are openly archived.
V. W. Couey, Catalog of Verified Mono-Monostatic Bodies (dataset). Zenodo, 2026. doi:10.5281/zenodo.20674394
| # | Member | Family | β | ECS | Stability gap | Mesh |
|---|---|---|---|---|---|---|
| 1 | Phase-1 (primary verified) | phase | 0.0231 | 1 | 2.58 × 10⁻⁵ | catalog_01_phase_beta0.023.stl |
| 2 | Phase-2 (sin(2*eta)) | phase | 0.0321 | 1 | 2.00 × 10⁻⁴ | catalog_02_phase2_beta0.032.stl |
| 3 | Phase-3 (sin(3*eta)) | phase | 0.0517 | 1 | 1.50 × 10⁻⁴ | catalog_03_phase3_beta0.052.stl |
| 4 | Radial-f3 (primary) | radial | 0.0231 | 1 | 7.53 × 10⁻⁵ | catalog_04_f3_beta0.023.stl |
| 5 | Radial-f4 (primary) | radial | 0.0231 | 1 | 5.40 × 10⁻⁵ | catalog_05_f4_beta0.023.stl |
| 6 | Radial-f3 scan | radial | 0.0231 | 1 | 1.41 × 10⁻⁴ | catalog_06_f3_beta0.023.stl |
| 7 | Radial-f4 scan | radial | 0.0231 | 1 | 1.29 × 10⁻⁴ | catalog_07_f4_beta0.023.stl |
| 8 | Radial-f4 low-beta | radial | 0.0150 | 1 | 8.25 × 10⁻⁶ | catalog_08_f4_beta0.015.stl |
| 9 | Radial-f4 high-beta | radial | 0.0350 | 1 | 2.58 × 10⁻⁴ | catalog_09_f4_beta0.035.stl |
| 10 | Radial-f3 low-beta | radial | 0.0150 | 1 | 5.93 × 10⁻⁵ | catalog_10_f3_beta0.015.stl |
| 11 | Min-asym beta=0.01 | radial | 0.0100 | 1 | 1.26 × 10⁻⁴ | catalog_11_f3_beta0.010.stl |
| 12 | Min-asym beta=0.008 | radial | 0.0080 | 1 | 8.98 × 10⁻⁵ | catalog_12_f3_beta0.008.stl |
| 13 | Min-asym beta=0.005 | radial | 0.0050 | 1 | 4.48 × 10⁻⁵ | catalog_13_f3_beta0.005.stl |
All 13 members confirmed mono-monostatic (13 of 13 at ECS = 1). Meshes are bundled in the archived dataset (doi:10.5281/zenodo.20674394).
How to read the catalog
Family is the construction route. Phase members perturb the generating curve's phase; radial members perturb its radius (with f3 / f4 denoting the harmonic order). The two families establish that mono-monostaticity here is not an artifact of a single construction but survives across independent parameterizations.
β is the perturbation magnitude — how far the shape departs from the rotationally symmetric base. ECS (Equilibrium Count Score) is the operational invariant: the number of stable equilibria; an ECS of 1 is the defining mono-monostatic property. The stability gap is the numerical margin by which the body avoids spurious additional equilibria — smaller gaps sit closer to the boundary of the mono-monostatic regime and are the more delicate constructions.
Provenance
The catalog is the empirical basis of the paper arXiv:2604.17120, which also documents a negative result: Sloan's analytical parameterization does not by itself yield mono-monostatic bodies. The verified members here were produced by the program's oracle stack and checked for equilibrium count; the complete meshes and landscape analysis are archived under doi:10.5281/zenodo.20674394 (CC-BY 4.0). For the methodology that underwrites the equilibrium verification, see methodology.